What is the mathematical basis for F-Curves?
After some digging through the Blender source code, I found the answer.

Short answer: They're just Bézier curves.

Long answer: They're normal Bézier curves with certain restrictions placed on the positions of the handles (the red circles). If the first handle is to the left of the second handle, as in the following image, then the curve is evaluated as a normal Bézier curve:



This guarantees that the curve will not turn back on itself.

However, if the handles cross (the first handle is to the right of the second handle), their positions need to be adjusted to avoid loops. The file fcurve.c (blender-2.77a\source\blender\blenkernel\intern\fcurve.c) gives the answer on line 2199:

/* adjust handles so that they don't overlap (forming a loop) */
correct_bezpart(v1, v2, v3, v4);


On line 1830, the correct_bezpart() function begins. The function finds the width of each handle and the width between the two keyframes. If the combined width of the two handles is greater than the width between the two keyframes, then the two handles are scaled down so that they don't cross each other, but their relative lengths are preserved. The original handles are used for drawing to the graph editor, but the scaled down versions are used for generating the curve, a plain-old cubic Bézier curve.

Here's an example F-Curve before correction. These handles are not used to create the actual curve; they're just drawn to the screen in Blender's graph editor:



Here's the same curve, but with corrected handles. These handles are not seen in the graph editor, but are the ones actually used to create the curve. If you want to pull out your ruler, you'll see that neither the lengths of each handle relative to each other, nor the angle of the handles were changed.



If you're not convinced, I invite you to look at fcurve.c and see for yourself. Overlapping handles are detected on lines 2188-2190. The actual handle correction takes place on lines 1830-1866. 

As far as why they're called "F-Curves", your guess is as good as mine. My guess is that the 'F' stands for 'function', meaning that, like a math function, each x-value maps to exactly one y-value (A Bézier curve is indeed a function, except that an (x, y) pair is a function of t).

Whether this method is the brainchild of Blender developers or was the result of some third party is still a mystery to me. The answer to that question, in my opinion, is more valuable than the method itself.